Concept of slope transcript:
Slide 1:Concept of slope
This presentation is about the concept of slope.
Slide 2: Connections
Slope is a fundamental mathematical concept that you meet in all five strands of mathematics. You
will meet several representations of slope and its important that you can recognise the concept
however it is represented.
Slide 3: A Physical Representation
You can have a physical representation of slope. The slope or gradient can be measured as a
measure of steepness. Consider these three lines, which do you think has the largest slope? In other
words which is the steepest?
Slide 4: A Geometric Representation
The slope can also be regarded as a geometric ratio of the rise, how far something goes up vertically
to the run, how far something goes out horizontally. Or the vertical change over the horizontal
change.
Slide 5: Rise over run
Can you see this. For the same run A rises most. Can you see the vertical change over the horizontal
change?
Slide 6: Visualising slope
Combining these two ideas leads to an algebraic representation. Slope m equals y2 minus y1 over x2
minus x1.
Slide 7: A trigonometric ratio
Slope m is equal to the rise over the run, which is equal to the change in the y value from A to B,
which is equal to the change in the x value from A to B which is y2 minus y1 over x2 minus x1. Here in
the diagram you can see that equals 4 take away 2 over 4 take away 2 which is equal to 2 over 2
which is one.
Slide 8: A Parametric Representation
The slope or gradient can be represented as the tan of the angle of inclination. You can examine this
trigonometric representation in the geometry and trigonometry presentation by clicking on the web
link.
Slide 9: A Calculus Concept
In general a linear equation is written in the form y = mx + c. Where m is the slope and y is where the
line cuts the x axis. Slope may also be regarded as a calculus concept.
Slide 10: Sample Question (1)
Calculus begins with the study of derivatives and rates of change. Slopes of lines help develop these
concepts.
Slide 11: Sample Question (2)
Look at the examination question and think about which representation is being used. What are you
given, what are you required to do?
Slide 12: Write down the equation of the line
Here you need to be able to compare slopes with l2 having the same slope as l1 and l3 having a
smaller slope and different intercept.
Slide 13: Higher Level task
What do you need to be able to do to write down the equation of the line shown? What information
can you get from the diagram? Draw the line with equation y = 2x – 4.
Slide 14: Lines sloping downwards
What do you know about these three lines? Could you sketch a diagram? You know their slopes,
what about their y intercepts? What about the x intercepts of these lines? Can you see these on the
diagram? Does this help you to write the equation of these three lines? Have a go.
Slide 15: Lines sloping downwards
Now what happens as the line slopes downwards as you go across to the right? How does the
downward direction effect the value of the slope?
Slide 16: Recognising slopes
Try to calculate the slope. Use your equation, can you see the rise? Can you see the run? What can
you say about the sign of the slope? You can find out more about positive and negative slopes by
clicking on the weblinks.
Slide 17: Predict..
This question requires you to recognise the slopes of the lines from their sketches. Have any of the
lines got negative slopes? How do you know? Have any of the lines got positive slopes? How do you
know? Check out the climbing analogy by clicking on the web link.
Slide 18: Put these in order
Plot each of the pairs of points, join them to form a line and predict whether the slope of each line
will be positive or negative. Check your predictions.
Slide 19: Identify slopes
Examine the lines, decide whether their slopes are positive or negative and then list the lines in
order of increasing slope.
Slide 20: Lines with the same steepness
Of the four lines pictured above, one has a slope of zero, one has a slope of one another has a slope
of negative one and another has an undefined slope. Decide which is which.
Slide 21: Perpendicular lines
Now think about lines which have exactly the same steepness, sketch two lines with the same
steepness. What can you say about these lines? Think about it will they ever meet? How will their
slopes compare now check your prediction. Write a statement about the slopes of parallel lines.
Slide 22: Some websites
Look at the two lines L1 and L2 they meet at an angle of 90°, this means they are perpendicular.
When two perpendicular lines meet there is a relationship between the two slopes. If you multiply
the two slopes, they will result in the answer of negative one. Think about why this would be true.
Slide 23: Question 1
By clicking on the weblinks you will get lots of practice in working with lines that are parallel or
perpendicular to each other.
Slide 24: Questions 2
This examination question requires to you to use these facts about parallel and perpendicular lines.
Slide 25: Questions 3
Look at the diagram, say whether the following statements is true or false and in each case give a
reason for the answer.
Slide 26: Question 3 continued
Look at the lines J, K, L, M, N what do you notice about them? Are there any lines parallel to each
other? What will that tell you about their slope? Are there any lines perpendicular to each other?
What will that tell you about their slope? Do any lines share y intercepts? Once you have had a
chance to look at the lines, look at the next slide and answer the questions.
Slide 27: Right angled triangles
Where else would you see lines that are perpendicular to each other? Right angled triangles.
Slide 28: Rectangles
Rectangles
Slide 29: Tangents to circles
Tangents to circles
Slide 30: Triangles inscribed in semi circles
Triangles inscribed in semi circles
Slide 31: Using slope to prove that lines meet at 90
Now you can prove that angles are 90° by using the concept of slope. If an angle is 90° then the two
lines that join to form them are perpendicular and the product of their slopes is negative 1.
Slide 32: Tangents, point of contact
Consider the following question. P is the point (0,7) and Q is the point (8, 11). Find the equation of
the circle with diameter PQ. Find the equation of the tangent at Q. Make a sketch, what do you
know about the tangent to a circle and the diameter or radius at the point of contact. Will this help
you solve the problem?
Slide 33:Question 4
Or this question. Two points A ( -3, 2) and B( 4, -1) are shown on the diagram below. Plot two
suitable points C and D so that ABCD is a parallelogram. Label their coordinates, by performing
suitable calculations, show that the figure you have drawn is a parallelogram.
Slide 34: Parallelogram
What are the properties of a parallelogram? Visit the website maths is fun to read about
parallelograms and their properties, how will these properties help you show that the parallelogram
ABCD you have drawn is a parallelogram. Will the slope help? Can you see any parallel lines? Can you
see any perpendicular lines? Have a go
Slide 35: Question 5
Now look at this question from a higher level sample paper. A circle has equation x2 + y2 – 2x + 4y –
15 =0 Find the value m for which the line m + 2y – 7 = 0 is a tangent to the circle. This is very similar
to a question we spoke of earlier. What do you know about the tangent and the diameter or radius
at the point of contact? Make a sketch.
Slide 36: Task 1
Try this plot the points (2, 3) and ( 2, 5) join these points to form a line L from the point (5, 4)
construct a line perpendicular to L. Show all the construction marks. Show using slopes that these
two lines are perpendicular to each other. What do you know about perpendicular lines?What do
you know about the slopes of these lines?have a go.
Slide 37: Other representations of slope
In the real world the graphical axes are not x and y but represent real life quantities such as velocity,
time, flow of electricity, productivity of farmlands, efficiency of assembly lines, cost of electricity
number of units used and many other quantities. Relationships between these quantities may
appear as stories or situations, tables graphs or formulae. Slope has meaning in each of these
representations. Most fields of study do graphical analysis of data. One of the most important way
to analyse a graph is to investigate it’s slope or rate of change.
Slide 38: Slope in real life
Consider the following real life situation. The cost of transporting document scan be represented by
the following straight line. Two points are given on the line A and B. What does A represent (0, 5)
what does the zero represent in this context. What does the 5 represent in this context? Similiarly B
is (6, 23) what does the 6 represent in this context? What does the 23 represent?
Slide 39: Calculations
You should be thinking that the point A represents zero on the distance axis and 5 on the cost axis.
So, when the documents are transported zero kilometres, the cost is 5 euro. And the point B
represents 6 on the distance axis and 23 on the cost axis. So when the documents are transported
6km the cost is 23 euro. Now you might be wondering why would anyone pay 5 euro to transport
documents zero kms? Well this is usually called a standing charge. So when you employ the courier
you’re bill is usually calculated with a standing charge and a charge for every kilometre. So how do
we calculate this amount for every km travelled? Well if you calculate the slope or gradient, that is
the change in the y axis over the change in the x axis and in this instance that’s the change in cost
over the change in distance, 23 – 5 over 6 – 0 which equals 18 euro over 6 km or three euro per
kilometre. So the rate charged by the courier is 3 euro per kilometre plus a standing charge of 5
euro. Can you see this on the graph? Can you see the standing charge? Where does it appear on the
graph? Can you see the 3 euro per kilometre how is this represented on the graph? What woud the
graph look like if it was 2 euro per km or 5 euro per km? Now take a look at this question.
Slide 40: Slope in real life
A cyclist cycles for 20 minutes at a constant speed and covers a distance of 15km as shown in the
diagram. Find the gradient of the line and describe it’s meaning in this context. What does the y
intercept mean in this instance? Look at the two graphs that show two varying quantities: cost and
number of units.
Slide 41: Example
Calculate the gradient of each line and attach a meaning to it. What is the meaning of the y intercept
in this context. Write an equation for both lines. Attach meaning to each equation.
Slide 42: Task 2
Have a look at the following real world situation.
You and your friends plan to visit the carnival next weekend. You don’t know how much it costs to
enter the carnival but you have the following incomplete information.
If you buy eight tickets the cost is 9 euro. If you buy 12 tickets, the cost is €11. It costs €12.50 to get
into the carnival but you don’t know how many tickets you get. So what are you asked to do? You’re
asked to write a linear equation in which y represents the total cost and x represents the number of
rides selected. You’re asked to identify the slope and y intercept in the equation and explain what
each of them means in the context of the problem. Then you have €20 to spend at the carnival. If
you need three tickets for each ride, how many rides will you be able to go on? Use maths to explain
your answer. Use words, symbols or both. Let’s try to be mathematical detectives: this is a linear
relationship. What does that mean? If you plot your data points and join them, what will the graph
look like? What does the y-intercept mean? What does slope mean in this situation? Have a go.
Slide 43: Line of best fit
In other real world situations varying quantities are not related in such an exact way. When you plot
points that represent experimental data obtained in science or statistics, you draw the straight line
that best represents those points. You draw by eye a straight line such that it represents the best fit.
With the distance of the plotted points from this line as small as possible. This may mean that the
line does not go through any of the plotted points and that some points will be above the line and
some below it.
Slide 44: Task 3
Try this real world example.
Slide 45: Question 6
Consider activity 5.1 on page 121 and the questions from pages 124 – 130 in the NCCA student
materials. You can download these at http://www.ncca.ie/en/Curriculum_and_Assessment/PostPrimary_Education/Project_Maths/Resources/Revised_LC_Student_Resources_Strand_1.pdf
These deal with real life situations where you have to decide the extent to which varying quantities
are related. Now assess your understanding by trying these examination style questions.
Slide 46: What’s slope got to do with it?
Lets summarise what we’ve been talking about in this presentation. Slope has meaning in formulae,
tables, physical story situations and is closely related to the concept of derivative. Awareness and
understanding of this will help you solve problems. The next five slides contain problem relating to
the concept of slope. Use the many representations of slope discussed in this presentation to help
solve problems.
Slide 47 – Question
Slide 48 – Question
Slide 49 – Question
Slide 50 – Question
Slide 51: Non-constant slope
So far we have looked at situations where the slope is constant. That is where it doesn’t change. Are
there any situations where the slope is not constant? What would these look like in a table, graph,
equation. Look at these graphs what would you say about the slope in each case? Is it constant or
changing? How do you know?
Slide 52: Slope of a tangent to a curve
The Geogebra file should help you see what is happening to the slope in this situation. Download
geogebra at the link shown it is a very powerful investigational tool for mathematics.
Slide 53: Geogebra
Can you see from the Geogebra investigation that the slope is different at every point along the
curve? The slope of the curve at any point is the same as that of the tangent line at that point. Does
this give you a clue as to how to find the slope of a curve? If someone said to you find the slope of
that curve you would have to say “at what point would you like me to find that slope” because the
slope is always changing.
Slide 54: Real life curves
But what type of situations produce graphs like this? Look at the staircase towers problem on the
slide. Would a graph of this table be linear how can you tell? What is different about this situation
and one that produces a linear graph? Draw the graph. What is happening to the slope of this graph?
Can you see the slope in the table? Can you explain why the graph looks the way it does?
Slide 55: Generalising
Listen to Pete as he explains how he generalised the relationship between the tower number and
the number of tiles.
Pete:
I knew the graph was going to be a curve because the number of tiles went up by a different amount
each time. To make a second tower I added two more tiles and to make the third tower I added
three more tiles. So the slope is not constant it went up by two and then by three and then by four.
When I looked at the change of the change it was constant, it went up by one each time and I knew
there was a quadratic relationship between the tower numbers and the number of tiles. This meant
that the relationship is y = ax2 + bx + c, all I had to do was find the a, b, c values. Three values so
three pieces of information. I chose the first three values in the table (1, 1), (2, 3), (3,6) and then I
used simultaneous equations to find the values of a, b, c, so I could generalise the relationship to say
the number of tiles equals half the tower number by the tower number plus one. So if the tower
number is ten there will be 55 tiles and that’s easy to check.
Slide 56: Generalising 2
Listen to Anne who is following the Ordinary level syllabus as she explains her strategy. Well first of
all I looked at the towers and I could see that it was like growing rectangles, but only half of it. So the
first one is half of 1 times 2, the second is half of 2 x 3, the third is half of a 3 x 4…so the nth one is
half of a n(n+1) so the nth term for this one is
. Verify Anne’s strategy by building the
pattern.
Slide 57 – Verifying
Slide 58: The Deal…
The Deal- what do you think of this deal? Make a table to show how much money I will get for each
day for the first ten days of the month. Make observations about the values in the table. What
would the graph look like would it be linear? How do you know? Make a graph to check your
prediction. What is happening to the slope of this graph? Can you see the slope in the table? What is
the difference between this graph and the one you drew for staircase towers? Which one goes up a
faster rate….how do you know?
Slide 59: Generalising 3
Listen to Peter again as he explains how he generalised the relationship between the number of days
and the amount of money.
I used the table to put the values into. It was a good way to see what type of relationship you have. I
knew it wasn’t linear because the amount of money it went up by each day was different and the
amount it went up by had a pattern in itself 2, 4, 8 it was getting bigger very fast and when I looked
at the differences between these differences they weren’t constant. It wasn’t a quadratic
relationship either. It must be exponential. Knowing this helped me with the generalisation. I put
another column to the table and wrote the number out in the way I got it like 2, 2 x 2 because to get
this number I doubled the two and then 2 x 2 x2 because I doubled the two by two to get this
number and so on. It made it really easy for me to generalise the relationship. The amount of money
my Dad would give me on the thirtieth day of the month would be 230 which is a huge amount of
money 1073,741,821 cent.
Slide 60: Classifying Relationships
Think about the relationships you have seen there are three main types. What does a linear
relationship look like in a context, in a table, in a graph, in a generalised expression? What does a
quadratic relationship look like? In a context, in a table, in a graph, in a generalised expression. What
does an exponential relationship look like in a context, table, graph, expression?
Slide 61: Think
What is an arithmetic sequence? Look it up in your textbook or ask your teacher or Google it.
Side 62: Searching for connections
The extract from Wikipedia defines an arithmetic sequence or arithmetic progression. Try to graph
this data. How would you represent it in a table? Now graph it, does it look like a linear relationship?
Can you see the slope in the table? In the sequence, generalise the linear relationship. Is this the
same as the nth term in the Arithmetic sequence.
Slide 63: Generalising
Generalise the linear relationship and use the arithmetic sequence formula to find the nth term of
the sequence in two different ways.
Slide 64: Examining student work
Examine this student’s work, can you see the slope of the linear relationship? What does this
represent in the arithmetic sequence? Look at the formula for the nth term of the arithmetic
sequence, compare this with how you generalise the linear relationship.
Slide 65: Think
Now think of Geometric sequences and think about how they compare with exponential
relationships. What is a geometrical sequence? Look at this extract from Wikipedia that defines a
geometric progression or geometric sequence.
Slide 66: And more connections..
Slide 67: Generalising
Find the nth term. Now generalise the exponential relationship.
Slide 68: Examining student work
Examine this student’s work. Compare each method, why did the student include the third column in
the table. What does the three represent?
Slide 69: Classifying relationships
You have now seen several ways to represent relationships between variables. What characterises a
linear relationship? What makes it different to an exponential or a quadratic relationship, how could
you determine a relationship between variables from a graph tables.
Slide 70: Providing evidence
Have a look at the understanding Financial Maths file, You will see how this work is applied in
financial situations. When you are done with that check out the making sense of the derivative file to
see how the concept of slope relates to differentiation and the derivative.
Slide 20
Of the lines pictured above, one has a slope of zero. One has a slope of one